Converting string to illegal/unsupported expression in Mathematica. - string

Question

I need to input a variable, say var, into Mathematica function Series[ ] like this: Series[A^2+B^2+C^2, var]. Series[ ] has the following syntax:
Series[f, {x, x_0, n}] generates a power series expansion for f about the point x=x_0 to order n.
Series[f, {x, x_0, n}, {y, y_0, m}, ...] successively finds series expansions with respect to x, then y, etc.
Because I am not always computing Series[ ] in one dimension (i.e., B and C are not always variables at each iteration), var must be properly formatted to fit the dimension demands. The caveat is that Mathematica likes lists, so any table degenerated will have a set of outer {}.
Suppose my previous code generates the following two sets of sets:
table[1]= {{A, 0, n}};
table[2]= {{A, 0, n}, {B, 0, m}}; .
My best idea is to use string manipulation (for i= 2):
string = ToString[table[i]]; .
str = StringReplacePart[string, {" ", " "}, {{1}, {StringLength[string], StringLength[string]}}]
The next step is to convert str to an expression like var and do Series[A^2 + B^2 + C^2, var] by doing var= ToExpression[str], but this returns the following error:
ToExpression::sntx: Invalid syntax in or before "{A, 0, n}, {B, 0, m}".
$Failed
Help convert str to expression propertly or suggest another way to handle this problem.

Answer 1

If I understood correctly, you have
table[2] = {{A, 0, n}, {B, 0, m}};
and are trying to obtain from that
Series[f[A,B],{A,0,n},{B,0,m}]
This may be done using Sequence, like so (I will use series instead of Series to keep it unevaluated so you can see what is happening):
series[f[A, B], Sequence ## table[2]]
(*
-> series[f[A,B],{A,0,n},{B,0,m}]
*)
So for instance
table[3] = {{A, 0, 2}, {B, 0, 2}};
Series[f[A, B], Sequence ## table[3]]
gives the right series expansion.

Answer 2

You can use First or Last or more generally, Part to get the List you want. For e.g.,
var = {{x, 0, 3}, {x, 0, 5}};
Series[1/(1 + x), var[[1]]]
Out[1]= 1 - x + x^2 - x^3 + O[x]^4
Series[1/(1 + x), var[[2]]]
Out[2]= 1 - x + x^2 - x^3 + x^4 - x^5 + O[x]^6
EDIT:
For multiple variables, you can use a SlotSequence (##) along with Apply (##) like so:
Series[Sin[u + w], ##] & ## {{u, 0, 3}, {w, 0, 3}}